Metamath Proof Explorer

Theorem lsmub2

Description: Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014) (Revised by Mario Carneiro, 19-Apr-2016)

		Ref	Expression
	Hypothesis	lsmub1.p	$⊢ \underset{˙}{\oplus} = LSSum (G)$
	Assertion	lsmub2	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G)) \to U \subseteq T \underset{˙}{\oplus} U$

Step	Hyp	Ref	Expression
1		lsmub1.p	$⊢ \underset{˙}{\oplus} = LSSum (G)$
2		subgsubm	$⊢ T \in SubGrp (G) \to T \in SubMnd (G)$
3		eqid	$⊢ {Base}_{G} = {Base}_{G}$
4	3	subgss	$⊢ U \in SubGrp (G) \to U \subseteq {Base}_{G}$
5	3 1	lsmub2x	$⊢ (T \in SubMnd (G) \land U \subseteq {Base}_{G}) \to U \subseteq T \underset{˙}{\oplus} U$
6	2 4 5	syl2an	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G)) \to U \subseteq T \underset{˙}{\oplus} U$